On Directed Covering and Domination Problems

Abstract

In this paper, we study covering and domination problems on directed graphs.
Although undirected Vertex Cover and Edge Dominating Set are well-studied classical graph problems, the directed versions have not been studied much due to the lack of clear definitions.
We give natural definitions for Directed r-In (Out) Vertex Cover and Directed (p,q)-Edge Dominating Set as directed generations of Vertex Cover and Edge Dominating Set.
For these problems, we show that
(1) Directed r-In (Out) Vertex Cover and Directed (p,q)-Edge Dominating Set are NP-complete on planar directed acyclic graphs except when r=1 or (p,q)=(0,0),
(2) if r>=2, Directed r-In (Out) Vertex Cover is W[2]-hard and (c*ln k)-inapproximable on directed acyclic graphs,
(3) if either p or q is greater than 1, Directed (p,q)-Edge Dominating Set is W[2]-hard and (c*ln k)-inapproximable on directed acyclic graphs,
(4) all problems can be solved in polynomial time on trees, and
(5) Directed (0,1),(1,0),(1,1)-Edge Dominating Set are fixed-parameter tractable in general graphs.
The first result implies that (directed) r-Dominating Set on directed line graphs is NP-complete even if r=1.